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Metamath Proof Explorer

Theorem cdleme0ex1N

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 9-Nov-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme0.l 
|- .<_ = ( le ` K )
cdleme0.j 
|- .\/ = ( join ` K )
cdleme0.m 
|- ./\ = ( meet ` K )
cdleme0.a 
|- A = ( Atoms ` K )
cdleme0.h 
|- H = ( LHyp ` K )
cdleme0.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme0ex1N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` K )
2 cdleme0.j 
 |-  .\/ = ( join ` K )
3 cdleme0.m 
 |-  ./\ = ( meet ` K )
4 cdleme0.a 
 |-  A = ( Atoms ` K )
5 cdleme0.h 
 |-  H = ( LHyp ` K )
6 cdleme0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
8 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
9 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> Q e. A )
10 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> P =/= Q )
11 1 2 3 4 5 6 lhpat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
12 7 8 9 10 11 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U e. A )
13 simp2ll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> P e. A )
14 1 2 3 4 5 6 cdlemeulpq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )
15 7 13 9 14 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U .<_ ( P .\/ Q ) )
16 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> K e. HL )
17 16 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> K e. Lat )
18 eqid 
 |-  ( Base ` K ) = ( Base ` K )
19 18 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
20 16 13 9 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) )
21 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> W e. H )
22 18 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> W e. ( Base ` K ) )
24 18 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
25 17 20 23 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
26 6 25 eqbrtrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> U .<_ W )
27 breq1 
 |-  ( u = U -> ( u .<_ ( P .\/ Q ) <-> U .<_ ( P .\/ Q ) ) )
28 breq1 
 |-  ( u = U -> ( u .<_ W <-> U .<_ W ) )
29 27 28 anbi12d 
 |-  ( u = U -> ( ( u .<_ ( P .\/ Q ) /\ u .<_ W ) <-> ( U .<_ ( P .\/ Q ) /\ U .<_ W ) ) )
30 29 rspcev 
 |-  ( ( U e. A /\ ( U .<_ ( P .\/ Q ) /\ U .<_ W ) ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )
31 12 15 26 30 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )